The Anderson impurity model (AIM) has long served as a cornerstone in the study of correlated electron systems. While numerical renormalization group (RG) offers great flexibility for metallic reservoirs, it becomes impossible in an unbiased way when a spectral gap $\mathrm{\Delta }$ opens up in the tunneling density of states. The only known exception is provided by the superconducting bath. In this paper, we lift these limitations by developing a numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches that accumulate at the gap edges. This reveals an unusual double scaling, which is a shared behavior between the superconducting and the scalar gapped AIMs. Moreover, it requires a special iterative diagonalization procedure with an alternating scheme for discarding states only every second iteration. The discretization and the diagonalization scheme form together what we refer to as the log-gap numerical RG. It is successfully applied to the superconducting and to the scalar gapped AIM. Consequently, it reveals that both models belong to the same RG equivalence class, which manifests physically in common singlet-doublet quantum phase transitions accompanied by in-gap bound states of given parities. While superconducting AIM is mainly used for benchmarking the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum. The gapped AIM is then tackled in an exact numerical RG approach and confirms quantitatively the assertions based on models with auxiliary metallic leads, but reveals that the other common approach used, for example, in the work of Moca and Roman [Phys. Rev. B 81, 235106 (2010)] is of strictly approximate nature.

• Received 17 July 2023
• Revised 25 September 2023
• Accepted 18 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.195123